ON THE MEAN SUMMABILITY BY CESARO METHOD OF FOURIER TRIGONOMETRIC SERIES IN TWO-WEIGHTED SETTING A GUVEN AND V KOKILASHVILI Received 26 June 2005; Revised 19 October 2005; Accepted 23 October 2005 The Cesaro summability of trigonometric Fourier series is investigated in the weighted Lebesgue spaces in a two-weight case, for one and two dimensions These results are applied to the prove of two-weighted Bernstein’s inequalities for trigonometric polynomials of one and two variables Copyright © 2006 A Guven and V Kokilashvili This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction It is well known that (see [9]) Cesaro means of 2π-periodic functions f ∈ L p (T) (1 ≤ p ≤ ∞) converges by norms Hereby T is denoted the interval (−π,π) The problem of the mean summability in weighted Lebesgue spaces has been investigated in [6] A 2π-periodic nonnegative integrable function w : T → R1 is called a weight funcp tion In the sequel by Lw (T), we denote the Banach function space of all measurable 2π-periodic functions f , for which f p,w = p T 1/ p f (x) w(x)dx < ∞ (1.1) In the paper [6] it has been done the complete characterization of that weights w, p for which Cesaro means converges to the initial function by the norm of Lw (T) Later on Muckenhoupt (see [3]) showed that the condition referred in [6] is equivalent to the condition A p , that is, sup |I | I w(x)dx |I | I w1− p (x)dx p −1 < ∞, (1.2) where p = p/(p − 1) and the supremum is taken over all one-dimensional intervals whose lengths are not greater than 2π Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 41837, Pages 1–15 DOI 10.1155/JIA/2006/41837 Mean summability of Fourier trigonometric series The problem of mean summability by linear methods of multiple Fourier trigonometp ric series in Lw (T) in the frame of A p classes has been studied in [5] In the present paper we investigate the situation when the weight w can be outside of A p class Precisely, we prove the necessary and sufficient condition for the pair of p weights (v,w) which governs the (C,α) summability in Lv (T) for arbitrary function f p from Lw (T) This result is applied to the prove of two-weighted Bernstein’s inequality for trigonometric polynomials It should be noted that for monotonic pairs of weights for (C,1) summability was studied in [7] Let ∞ f (x) ∼ a0 an cosnx + bn sinnx + n =1 (1.3) be the Fourier series of function f ∈ L1 (T) Let α σn (x, f ) = π π −π α f (x + t)Kn (t)dt, α>0 (1.4) when α −1 An−k Dk (t) , Aα n k =0 n α Kn = (1.5) with k Dk (t) = sin(ν + 1/2)t , 2sin(1/2)t ν =0 (1.6) n+α nα ≈ Aα = n α Γ(α + 1) In the sequel we will need the following well-known estimates for Cesaro kernel (see [9, pages 94–95]): α Kn (t) ≤ 2n, α Kn (t) ≤ cα n−α |t |−(α+1) (1.7) when < |t | < π Two-weight boundedness and mean summability (one-dimensional case) Let us introduce the certain class of pairs of weight functions Definition 2.1 A pair of weights (v,w) is said to be of class Ꮽ p (T), if sup |I | I v(x)dx |I | I w1− p (x)dx p −1 < ∞, (2.1) where the least upper bound is taken over all one-dimensional intervals by lengths not more than 2π A Guven and V Kokilashvili The following statement is true Theorem 2.2 Let < p < ∞ Then α lim σn (·, f ) − f p,v n→∞ =0 (2.2) p for arbitrary f from Lw (T) if and only if (v,w) ∈ Ꮽ p (T) The proof is based on the following statement Theorem 2.3 Let < p < ∞ For the validity of the inequality α σn (·, f ) p,v ≤c f (2.3) p,w p for arbitrary f ∈ Lw (T), where the constant c does not depend on n and f , it is necessary and sufficient that (v,w) ∈ Ꮽ p (T) Note that the condition (v,w) ∈ Ꮽ p (T) is also necessary and sufficient for boundedness of p p the Abel-Poisson means from Lw (T) to Lv (T) [4] First of all let us prove two-weighted inequality for the average x+h h1−β x −h β fh (x) = h > 0, ≤ β < f (t) dt, (2.4) The last functions are an extension of Steklov means Theorem 2.4 Let < p < q < ∞ and let 1/q = 1/ p − β If the condition sup I |I | 1/q I v(x)dx |I | I w1− p (x)dx 1/ p the following inequality holds: π β −π 1/q q fh (x) v(x)dx π ≤c −π p f (x) w(x)dx 1/ p (2.6) Proof Let h ≤ π and N be the least natural number for which Nh ≥ π Then we have β T q fh (x) v(x)dx N −1 ≤ k=−N kh N −1 ≤ (k+1)h (k+1)h k=−N kh h−q(1−β) h−q(1−β) q x+h x −h f (t) dt v(x)dx (k+2)h (k−1)h q f (t) dt v(x)dx Mean summability of Fourier trigonometric series N −1 ≤ (k+1)h k=−N kh N −1 kh k=−N (k+2)h × N −1 k=−N (k+2)h v(x)dx (k−1)h w1− p (t)dt q/ p (k−1)h w1− p (t)dt q/ p v(x)dx h−q(1−β) q/ p p (k+1)h h v(x)dx kh (k+2)h f (t) w(t)dt (k−1)h h q/ p p f (t) w(t)dt (k−1)h (k+1)h = = (k+2)h h−q(1−β) (k+2)h (k−1)h q/ p w1− p (t)dt (k+2)h p f (t) w(t)dt (k−1)h q/ p (2.7) Arguing to the condition (2.5) we conclude that π −π β N −1 q fh (x) v(x)dx ≤ c k=−N (k+2)h (k−1)h q/ p p f (t) w(t)dt (2.8) Using [2, Proposition 5.1.3] we obtain that π β q fh (x) v(x)dx ≤ c1 f −π q p,w (2.9) Theorem is proved Note that Theorem 2.4 is proved in [4] in the case β = Proof of Theorem 2.3 Let us show that α σn (x, f ) ≤ c0 2π −1−α h fh (x)dh, α 1/n n (2.10) where the constant c0 does not depend on f and h By reversing the order of integration in the right side integral of (2.10), we get that it is more than or equal to I= ≥c x+π x −π 2π f (t) x+π x −π −2−α h dh dt α max(|x−t |,1/n) n 1 f (t) α max |x − t |, n n (2.11) −1−α dt since |x − t | ≤ π Indeed, let us show that for |x − t | ≤ π, the inequality 2π max{|x−t |,1/n} h−2−α dh > c max |x − t |,1/n where c does not depend on x, t, and n −α −1 , (2.12) A Guven and V Kokilashvili It is obvious that I1 = 2π max{|x−t |,1/n} h−2−α dh = 1+α max |x − t |,1/n 1+α − (2π)1+α (2.13) To prove the latter inequality we consider two cases (a) Let |x − t | < 1/n Then I1 = 1 n1+α − > − (2π)−1−α n1+α 1+α (2π)1+α 1+α (2.14) (b) Let now |x − t | ≥ 1/n Then for the sake of the fact |x − t | ≤ π, we conclude that I1 = 1 1 1 − + − = + α |x − t |1+α (2π)1+α 2(1 + α) |x − t |1+α |x − t |1+α (2π)1+α > 1 1 1 ≥ + − + − 2(1 + α) |x − t |1+α π 1+α (2π)1+α 2(1 + α) |x − t |1+α π 1+α 2α π 1+α > 1 2(1 + α) |x − t |1+α (2.15) which implies the desired result Using the estimates (1.7) we obtain that I ≥c x+π x −π π α f (t) Kn (x − t)dt ≥ c −π α α f (t)Kn (x − t)dt = c σn (x, f ) (2.16) Thus we obtain (2.10) Passing to the norms in (2.10), then applying Theorem 2.4 by Minkowski’s integral inequality we obtain that p T p α σn (x, f ) v(x)dx ≤ c T ≤ c1 f (x) w(x) nα 1/n h−1−α dh p dx (2.17) p T f (x) w(x)dx Now we will prove that from (2.3) it follows that (v,w) ∈ Ꮽ p (T) If the length of the interval I is more than π/4, the validness of the condition (2.1) is clear Let now |I | ≤ π/4 Let m be the greatest integer for which m≤ π − 2|I | (2.18) Then we have k+ π (x − t) ≤ (m + 1)|x − t | ≤ 2 (2.19) Mean summability of Fourier trigonometric series Then applying Abel’s transform we get that for x and t from I, the following estimates are true: m α Km (x − t) ≥ m Aα −k m (2k + 1) ≥ c(m + 2) Aα−1 (k + 1) Aα (m + 1)Aα k=0 m−k m m k =0 (2.20) m Aα+1 c c c m ≥ Aα −k = ≥ m α |I | (m + 1)Am k=0 |I | (m + 1)Aα |I | m Let us put in (2.3) the function f0 (x) = w1− p (x)χI (x) (2.21) for m which was indicated above Then we obtain I I p α w1− p (t)Km (x − t)dt v(x)dx ≤ c I w1− p (x)dx (2.22) From the last inequality by (2.20) we conclude that I |I | I w1− p (t)dt p v(x)dx ≤ c I w1− p (x)dx (2.23) Thus from (2.3) it follows that (v,w) ∈ Ꮽ p (T) Proof of Theorem 2.2 Let us show that if (v,w) ∈ Ꮽ p (T), then α lim σn (·, f ) − f n→∞ p,v =0 (2.24) p for arbitrary f ∈ Lw (T) Consider the sequence of linear operators: α Un : f −→ σn ·, f p (2.25) p It is easy to see that Un is bounded from Lw (T) to Lv (T) Indeed applying Hă lders ino equality we get p p T α σn (x, f ) v(x)dx ≤ 2n ≤ 2n T T f (t) dt v(x)dx p f (t) w(t)dt T T v(x)dx T w 1− p (2.26) p −1 (x)dx By our assumptions all these integrals are finite, the constant c = 2n does not depend on f T v(x)dx T w1− p (x)dx p −1 (2.27) A Guven and V Kokilashvili Then since (v,w) ∈ Ꮽ p (T) by Theorem 2.3, we have that the sequence of operators norms is bounded On the other hand, the set of all 2π-periodic continuous on the line p functions is dense in Lw (T) It is known (see [9]) that the Cesaro means of continuous function uniformly converges to the initial function and since v ∈ L1 (T) they converge p in Lv (T) as well Applying the Banach-Steinhaus theorem (see, [1]) we conclude that the p convergence holds for arbitrary f ∈ Lw (T) p Now we prove the necessity part From the convergence in Lv (T) of the Cesaro means by Banach-Steinhaus theorem we conclude that Un ∞ p (2.28) p Lw (T)→Lv (T) n=1 is bounded It means that (2.3) holds Then by Theorem 2.3 we conclude that (v,w) ∈ Ꮽ p (T) Theorem is proved On the mean (C, α, β) summability of the double trigonometric Fourier series Let T2 = T × T and f (x, y) be an integrable function on T2 which is 2π-periodic with respect to each variable Let ∞ f (x, y) ∼ λmn amn cosmx cosny + bmn sinmx sinmy m,n=0 (3.1) + cmn cosmx sinny + dmn sinmx sinny , where ⎧ ⎪1 ⎪ , ⎪ ⎪4 ⎪ ⎪ ⎨ when m = n = 0, λmn = ⎪ , for m = 0, n > or m > 0, n = 0, ⎪2 ⎪ ⎪ ⎪ ⎪ ⎩1, when m > 0, n > (3.2) Let n α −1 β −1 j =0 Am−i An− j Si j (x, y, β Aα An m m i =0 (α,β) σmn (x, y, f ) = f) (α,β > 0) , (3.3) be the Cesaro means for the function f , where Si j (x, y, f ) are partial sums of (3.1) We consider the mean summability in weighted space defined by the norm f p,w = p T2 f (x, y) w(x, y)dx d y 1/ p , where w is a weight function of two variables In this section our goal is to prove the following result and some its converse (3.4) Mean summability of Fourier trigonometric series Theorem 3.1 Let < p < ∞ Assume that the pair of weights (v,w) satisfies the condition sup J |J | J |J | v(x, y)dx d y p −1 w1− p (x, y)dx d y J < ∞, (3.5) where the least upper bound is taken over all rectangles, with the sides parallel to the coordip nate axes Then for arbitrary f ∈ Lw (T2 ), we have (α,β) lim σmn (·, ·, f ) − f m→∞ n→∞ p,v − → (3.6) In the sequel the set of all pairs with the condition (3.5) will be denoted by Ꮽ p (T2 , J) Here J denotes the set of all rectangles with parallel to the coordinate axes The proof of this theorem is based on the following statement Theorem 3.2 Let < p < ∞ and (v,w) ∈ Ꮽ p (T2 , J), then (α,β) σmn (·, ·, f ) p,v ≤c f p,w , (3.7) with the constant c independent of m, n, and f To prove Theorem 3.2 we need the two-dimensional version of Theorem 2.4 Let us consider generalized multiple Steklov means γ fhk (x) = sup h>0 k>0 (hk)γ x+h y+k x −h y −k f (t,τ) dt dτ, < γ ≤ (3.8) Theorem 3.3 Let < p < ∞ and 1/q = 1/ p − γ Let (v,w) ∈ Ꮽ p (T2 , J) Then there exists p a constant c > such that for arbitrary f ∈ Lw (T2 ) and positive h and k, we have γ fhk q,v ≤c f p,w (3.9) Proof Let h ≤ π and k ≤ π Let M and N be the least natural numbers for which Mh ≥ π and Nk ≥ π Then γ T2 M q fhk (x, y) v(x, y)dx d y ≤ N i=−M j =−N × x+h x −h M −1 N −1 ≤ (i+1)h ( j+1)k ih jk q y+k y −k f (t,τ) dtdτ (i+1)h i=−M j =−N ih × (i+2)h (i−1)h (hk)−q(1−γ) (3.10) ( j+1)k jk v(x, y)dx d y (hk) −q(1−γ) q ( j+1)k ( j −1)k f (t,τ) dtdτ v(x, y)dx d y A Guven and V Kokilashvili Using the Hă lders inequality we get o γ T2 q fhk (x, y) v(x, y)dx d y M −1 N −1 ih i=−M j =−N jk (i+2)h ( j+2)k (i−1)h × ( j+1)k (i+1)h ≤ ( j −1)k (hk)−q(1−γ) w1− p (x, y)dx d y (i+2)h (i−1)h ( j+1)k q/ p p f (t,τ) w(t,τ)dtdτ ( j −1)k q/ p v(x, y)dx d y (3.11) By the condition Ꮽ p (T2 , J) we derive that γ T2 M −1 N −1 q fhk (x, y) v(x, y)dx d y ≤ c i=−M j =−N (i+2)h (i−1)h ( j+1)k ( j −1)k q/ p p f (t,τ) w(t,τ)dtdτ (3.12) Consequently, γ T2 q fhk (x, y) v(x, y)dx d y ≤ c f q p,w (3.13) Theorem is proved Proof of Theorem 3.2 Let us prove that (α,β) σmn (x, y, f ) ≤ c π π 1/m 1/n −1−α −1−β h k fhk (x, y, f )dhdk, mα n β (3.14) where the constant does not depend on f , x, y, m, and n If we reverse the order of integration in right side of (3.14), then by the arguments similar to that of the one-dimensional case we obtain that I= ≥c x+π x −π x −π x+π y+π y −π 2π y+π y −π f (t,s) 2π max(|x−t |,1/m) f (t,s) max(| y −s|,1/n) 1 max |x − t |, α nβ m m −2−α −2−β h k dhdk dt ds mα n β −1−α max | y − s|, n −1−β dt ds (3.15) Applying the known estimates for Cesaro kernel from the last estimate we derive that I ≥c We proved (3.14) β T2 (α,β) α f (t,s) Km (x − t)Kn (y − s)dt ds ≥ c σmn (x, y, f ) (3.16) 10 Mean summability of Fourier trigonometric series Taking the norms in (3.14), by Theorem 3.3 and Minkowski’s inequality we conclude that p (α,β) T2 σmn (x, y, f ) v(x, y)d dx d y ≤c ≤ c1 p T2 f (x, y) w(x, y) 2π 2π 1/m α nβ m 1/n h−1−α k−1−β dhdk p dx d y (3.17) p T2 f (x, y) w(x, y)dx d y By this we obtain (3.7) Proof of Theorem 3.1 Consider the sequence of operators (α,β) Umn : f −→ σmn (·, ·, f ) (3.18) It is evident that Umn is linear bounded for each (m,n) as T2 v(x, y)dx d y < ∞, T2 w1− p (x, y)dx d y < ∞ (3.19) Then since (v,w) ∈ Ꮽ p (T2 , J) by Theorem 3.2, the sequence of operators norms Umn p p ∞ (3.20) Lw →Lv m,n=1 is bounded On the other hand, the set of 2π-periodic functions which are continuous on p the plane is dense in Lw (T2 ) Then it is known that Cesaro means of Lipschitz functions of two variables converges uniformly (see [8, page 181]) Since v ∈ L1 (T2 ) the last converp gence we have by means of Lv norms as well Applying the Banach-Steinhaus theorem (see p [1]) we conclude that the norm convergence (3.6) holds for arbitrary f ∈ Lw (T2 ) Theorem 3.4 Let < p < ∞ If the inequality (3.7) is satisfied, then the condition (3.5) holds when the least upper bound is taken over all rectangles J0 = I1 × I2 and |I1 | < π/4 and |I2 | < π/4 Proof Let m and n be that greatest natural numbers with π π ≤ I1 ≤ , 2(m + 2) 2(m + 1) π π ≤ I2 ≤ 2(n + 2) 2(n + 1) (3.21) Then for (x, y) ∈ J0 and (t,τ) ∈ J0 , we have α Km (x − t) ≥ c |I1 | , β Kn (y − s) ≥ c |I2 | with some constant c nondepending on m, n, (x, y) and (t,s) (3.22) A Guven and V Kokilashvili 11 α Indeed Abel’s transform for Km gives m Aα −k m (2k + 1) ≥ c(m + 2) Aα−1 (k + 1) Aα (m + 1)Aα k=0 m−k m m k =0 m α Km (x − t) ≥ n Aα+1 c c c m ≥ Aα = ≥ , k α |I1 | (m + 1)Am k=0 |I1 | (m + 1)Aα |I1 | m (3.23) for (x, y) ∈ J0 and (t,s) ∈ J0 β Analogously we can estimate Kn (y − s) Now for indicated m and n, put (3.7) in the function f0 (x, y) = w1− p (x, y)χJ0 (x, y) (3.24) Then we get β J0 J0 α w1− p (t,s)Km (x − t)Kn (y − s)dt ds p v(x, y)dx d y ≤ c J0 w1− p (x, y)dx d y (3.25) By (3.23) from the last inequality we obtain J0 |J0 | p J0 w 1− p (t,s)dt ds v(x, y)dx d y ≤ c J0 w1− p (x, y)dx d y, (3.26) which is (3.5) with the least upper bound taken over all rectangles J0 , such that J0 = I1 × I2 and |Ii | < π/4, i = 1,2 Theorem 3.5 Let < p < ∞ If (3.7) holds, then there exist k ∈ N and a positive c > such that |J | J v(x, y)dx d y |J | J w1− p (x, y)dx d y p −1